Meets Thursdays 11:15am-12:15am in Vincent 570 and/or this Zoom link.
Organizers: Ayah Almousa, Daoji Huang, Monica Lewis, and Michael Perlman.
Click here to add the seminar schedule to your Google Calendar.
Bring your lunch and meet outside of Vincent Hall at noon to get to know the rest of the commutative algebra group and discuss potential speakers for the semester. You can also join lunch over Zoom if you prefer.
We explore generic tropical hyperplane arrangements where some of the apices of the tropical hyperplanes are "taken to infinity''. We show that the resulting bounded complex gives rise to a cellular resolution for an ideal that is Alexander dual to the Stanley-Reisner ideal of a regular triangulation of a (type A) root polytope. Moreover, the Stanley-Reisner ideal of this triangulation can be seen as a squarefree initial ideal of a toric edge ideal of a bipartite graph; this key observation yields a new approach to studying homological aspects of toric edge ideals of bipartite graphs. This is joint work with Anton Dochtermann (Texas State) and Benjamin Smith (Manchester).
We consider a smooth complex variety endowed with the action of a linear algebraic group, such as a toric variety, space of matrices, or flag variety. Given an orbit, the local cohomology modules with support in its closure encode a great deal of information about its singularities and topology. We will discuss how techniques from representation theory, D-modules, and quivers may be used to compute these local cohomology modules and their related invariants.
The local cohomology of a regular ring is known in many cases to exhibit remarkable finiteness properties. If R is a polynomial ring over a field of characteristic 0, the local cohomology of R is finitely generated (in fact, holonomic) over the ring of differential operators. In characteristic p > 0, the local cohomology of R is finitely generated (in fact, unit) over the ring R<F>, defined by the relation r^pF = Fr for r in R. Much of this structure is unavailable if R has singularities, but in some circumstances, enough useful structure remains to enable the proof of powerful finiteness results. In this talk, we will review the prime characteristic setting, with a focus on the sets of associated and minimal primes of local cohomology, and will present joint work with Eric Canton on the case of complete intersection rings.
Motivated by toric geometry, Maclagan-Smith defined the multigraded Castelnuovo-Mumford regularity for sheaves on a simplicial toric variety. While this definition reduces to the usual definition on a projective space, other descriptions of regularity in terms of the Betti numbers, local cohomology, or resolutions of truncations of the corresponding graded module proven by Eisenbud and Goto are no longer equivalent. I will discuss recent joint work with Lauren Cranton Heller and Juliette Bruce on generalizing Eisenbud-Goto's conditions to the "easiest difficult" case, namely products of projective spaces, and our hopes and dreams for how to do the same for other toric varieties.
Algebraic de Rham homology is a numerical invariant of an algebraic variety defined using algebraic differential forms. Hartshorne defined the de Rham homology of a closed subscheme Y of a smooth scheme X over a field k of characteristic zero as the hypercohomology of the de Rham complex on X with supports in Y. I will define the hypercohomology spectral sequence abutting to algebraic de Rham homology and discuss some results about its degeneration.
Differential modules are a generalization of chain complexes at least dating back to Cartan and Eilenberg, and have recently become an important framework for studying questions related to the ranks of syzygies and extensions of the BGG correspondence to non-standard gradings. Many properties that we take for granted in the case of modules and complexes fail for differential modules, but recent results of Brown and Erman indicate that minimal free resolutions still play an important role in the study of homological properties of differential modules. In this talk, we will introduce a "Koszul differential module", which generalizes the classical Koszul complex and serves as a free flag resolution for certain classes of differential modules with complete intersection homology. This is joint work with Maya Banks.
A flag variety admits an affine paving by Schubert cells, and each Schubert cell has a stratification by Kazhdan-Lusztig varieties, which is obtained by intersecting the Schubert cell with opposite Schubert varieties. This stratification enjoys many desirable properties. For example, every open stratum is smooth, and every closed stratum is normal, Cohen-Macaulay, and has rational singularities. I will talk about how to explicitly parametrize the Schubert cells with Bott-Samelson coordinates and compute equations for Kazhdan-Lusztig varieties in fintie type A using Fulton's matrix Schubert varieties. Using similar techniques, we compute equations for Kazhdan-Lusztig varieties in the affine type A flag variety. This involves introducing an analogue of Fulton's matrix Schubert varieties in the affine type A.
Toric face rings, introduced by Stanley, are simultaneous generalizations of Stanley--Reisner rings and affine semigroup rings, among others. We use the combinatorics of the fan underlying these rings to inductively compute their rings of differential operators. Along the way, we discover a new differential characterization of the Gorenstein property for affine semigroup rings. This is joint work with C-Y. Jean Chan, Patricia Klein, Laura Matusevich, Janet Page, and Janet Vassilev.
For a smooth surface S the Hilbert scheme of points S^(n) is a well studied smooth parameter space. In this talk I will consider a natural generalization, the nested Hilbert scheme of points S^(n,m) which parameterizes pairs of 0-dimensional subschemes X \supseteq Y of S with deg(X) = n and deg(Y) = m. In contrast to the usual Hilbert scheme of points, S^(n,m) is almost always singular and it is known that S(n,1) has rational singularities. I will discuss some general techniques to study S^(n,m) and apply them to show that S^(n,2) also has rational singularities. This relies on a connection between S^(n,2) and a certain variety of matrices, and involves square-free Gröbner degenerations as well as the Kempf-Weyman geometric technique. This is joint work with Alessio Sammartano.
Orbital varieties, first studied by Joseph in the context of the representation theory of symmetric groups, are certain subvarieties of upper triangular nilpotent matrices. The inclusion order on orbital varieties induces a partial order (the geometric order) on standard Young tableaux. A combinatorial description of the geometric order remains elusive in all but a few cases, which I will describe in this talk. I will also discuss the relationship with Schubert calculus, and recent progress and conjectures towards a description of the geometric order.
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We study asymptotic homological properties of random quadratic monomial ideals in a polynomial ring R = k[x_1, . . . , x_n], utilizing methods from the Erdos-Renyi model of random graphs. Here we consider a graph on n vertices and exclude an edge (corresponding to a quadratic generator of the ideal I) with probability p, and consider algebraic properties as n tends to infinity. Our main results involve fixing the edge parameter p = p(n) so that asymptotically almost surely the Krull dimension of R/I is fixed. Under these conditions we establish various properties regarding the Betti table of R/I, including sharp bounds on regularity and projective dimension and distribution of nonzero Betti numbers. These results extend work of Erman-Yang, who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Our results use collapsibility properties of random clique complexes and Garland’s method regarding spectral gaps of graphs, and in particular rely on the underlying field in some cases. This is joint work with Andrew Newman.